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3 years ago

2p2 -15p + 25 = (p - 5)(2p - 5) (p - 5)(2p + 5) (p + 5)(2p - 5) (2p - 15)(p + 5)

Mathematics

2 answers:

Anarel [89]3 years ago

7 0

I think it's the first one

aliina [53]3 years ago

3 0

(p - 5)(2p - 5), in the first option

<h3>Further explanation</h3>

The Problem:

Which are the factors of 2p² - 15p + 25 = ?

(p - 5)(2p - 5)
(p - 5)(2p + 5)
(p + 5)(2p - 5)
(2p - 15)(p + 5)

The Process:

The problem we have this time is how to solve factoring a quadratic equation.

First Way

2p² - 15p + 25 = 2p² - 10p - 5p + 25

= 2p(p - 5) - 5(p - 5)

= (2p - 5)(p - 5) or (p - 5)(2p - 5)

Recall this: $\boxed{ \ a(p + q) + b(p + q) \rightleftharpoons (a + b)(p + q) \ }$

Second Way

Allow me to introduce the "temporary quadratic" method. This is a unique way! Look carefully below.

2p² - 15p + 25 ⇒ multiply 2 as the coefficient of the p² term with 25 as a constant.

2 x 25 = 50.

Then, find two factors of 50 which if added are -15 as the coefficients of the term p.

We get the following: - 5 x - 10 = 50, i.e., -5 and -10.

Both are positioned in factor brackets. Simultaneously we place the term of 2p in both brackets as a sum operation with both factors. This is properly the uniqueness of this method!

2p² - 15p + 25 = "(2p - 10)(2p - 5)"

Because we call it the "temporary quadratic" method, simplify the factors in parentheses if it can still be done.

Thus, we get 2p² - 15p + 25 = (p - 5)(2p - 5), i.e. the two terms of (2p - 10) divided by 2.

Third Way

Let us check one by one of the answer options available.

(p - 5)(2p - 5) = p(2p - 5) - 5(2p - 5)

= 2p² - 5p - 10p + 25

= 2p² - 15p + 25 (yes, this is correct)

(p - 5)(2p + 5) = p(2p - 5) - 5(2p + 5)

= 2p² - 5p - 10p - 25

= 2p² - 15p - 25

(p + 5)(2p - 5) = p(2p - 5) + 5(2p - 5)

= 2p² - 5p + 10p + 25

= 2p² + 5p + 25

(2p - 15)(p + 5) = 2p(p + 5) - 15(p + 5)

= 2p² + 10p - 15p - 75

= 2p² - 5p - 75

_ _ _ _ _ _ _ _ _ _

Take a look at this example:

3m² - 10m - 8 = ?

= 3m² - 12m + 2m - 8
= 3m(m - 4) + 2(m - 4)
This becomes (3m + 2)(m - 4)

3m² - 10m - 8 = ?

3 x (-8) = -24
-24 = 2 x (-12), because 2 + (-12) = -10
3m² - 10m - 8 = "(3m + 2)(3m - 12)"
This becomes (3m + 2)(m - 4)

Notes:

$\boxed{ \ ax^2 + bx + c = 0, \ \ \ \ \ a \neq 0 \ }$ is a quadratic equation in standard form.

x is a variable
a, b, and c are real number coefficients
the only c is called a constant (no variables)

Methods of solving quadratic equations include:

Factoring and using the zero product property:
Using the square root property: $\boxed{ \ x^2 = a, \ then \ x = \pm \sqrt{a} \ }$
Completing the square: $\boxed{ \ x^2 + bx + \bigg({\frac{b}{2}} \bigg)^2 = \bigg( x + \frac{b}{2} \bigg)^2 \ }$
Using the quadratic formula: $\boxed{ \ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \ }$